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Laurent NottaleCNRS

LUTH, Observatoire de Paris-Meudon

http://www.luth.obspm.fr/~luthier/nottale/

Paris, ENS, October 8, EDU-2008

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Scales in naturePlanck scale10 cm-33

10 cm-28

10 cm-16

3 10 cm-13

4 10 cm-11

1 Angstrom

40 microns

1 m

6000 km700000 km1 millard km

1 parsec

10 10

10 20

10 30

10 40

10 50

10 60

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Grand Unification

accelerators: today's limitelectroweak unification

electron Compton lengthBohr radius

quarks

virus bacteries

human scale

Earth radiusSun radiusSolar System

distances to StarsMilky Way radius10 kpc

1 Mpc100 Mpc

Clusters of galaxiesvery large structuresCosmological scale10 cm28

atoms

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RELATIVITY

COVARIANCE EQUIVALENCE

weak / strong

Action Geodesical

CONSERVATIONNoether

FIRST PRINCIPLES

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Giving up the hypothesis of differentiability of

space-time

Explicit dependence of coordinates in terms of scale variables

+ divergence --> (theory : = dX ;experiment : = apparatus resolution)

Generalize relativity of motion ?

Transformations of non-differentiable coordinates ? ….

Theorem

FRACTAL SPACE-TIME

Complete laws of physics by fundamental scale laws

Continuity +SCALE RELATIVITY

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Principle of scale relativity

Scale covarianceGeneralized principle

of equivalence

Linear scale-laws:  “Galilean”self-similarity,

constant fractal dimension,scale invariance

Linear scale-laws :  “Lorentzian”varying fractal dimension,

scale covariance,invariant limiting scales

Non-linear scale-laws:  general scale-relativity,

scale dynamics,gauge fields

Constrain the new scale laws…

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A

A

0

1X

t0 1

1. Continuity + nondifferentiability Scale dependence

0.01 0.11

Continuity + Non-differentiability implies Fractality

when

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Continuity + Non-differentiability implies Fractality

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Continuity + Non-differentiability implies Fractality

divergence

Lebesgue theorem (1903):«  a curve of finite length is almost everywhere differentiable »

Since F is continuous and no where or almost no where differentiable

i.e., F is a fractal curve

2. Continuity + nondifferentiability

when

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*Re-definition of space-time resolution intervals as characterizing the state of scale of the coordinate system

*Relative character of the « resolutions » (scale-variables):only scale ratios do have a physical meaning, never an absolute scale

*Principle of scale relativity: « the fundamental laws of nature are valid in any coordinate system, whatever its state of scale  »

*Principle of scale covariance: the equations of physics keep their form (the simplest possible)* in the scale transformations

of the coordinate system

Weak: same form under generalized transformations

Strong: Galilean form (vacuum, inertial motion)

Principle of relativity of scales

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Origin

Orientation

Motion

Velocity

AccelerationScale

Resolution

Coordinate system

x

t

δ x

δ t

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FRACTALSFRACTALS

From fractal objectsFrom fractal objects

toto

Fractal space-timesFractal space-times

http://www.luth.obspm.fr/~luthier/nottale/

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Discrete zooms on a Discrete zooms on a fractal curvefractal curve

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von Koch von Koch curvecurve

F0

F1

F2

F3

F4

F∞

L0

L1 = L0 (p/q)

L2 = L0 (p/q)2

L3 = L0 (p/q)3

L4 = L0 (p/q)4

L∞ = L0 (p/q)∞

Generator:p = 4q = 3

Fractal dimension:

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Continuous zoom on a fractal Continuous zoom on a fractal curvecurve

Animation

QuickTime™ et undécompresseur Graphiquessont requis pour visionner cette image.

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Fractal geometry: space of positions and scales

© L. Nottale CNRS Observatoire de Paris-Meudon

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Curves of variable fractal dimension (in space)

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18

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QuickTime™ et undécompresseur Animationsont requis pour visionner cette image.

Animation

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Laws of transformation of the scale variables

From scale invariance to scale covariance

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Dilatation operator (Gell-Mann-Lévy method):

First order scale differential First order scale differential equation:equation:

Taylor expansion:

Solution: fractal of constant dimension + transition:

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ln L

ln ε

transitionfractal

scale -independent

ln ε

transitionfractal

delta

variation of the length variation of the scale dimension

"scale inertia"scale -independent

Case of « scale-inertial » laws (which are solutions of a first order

scale differential equation in scale space).

Dependence on scale of the length (=fractal coordinate)Dependence on scale of the length (=fractal coordinate) and of the effective fractal dimension and of the effective fractal dimension

= DF - DT

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Asymptotic behavior:

Scale transformation:

Law of composition of dilatations:

Result: mathematical structure of a Galileo group ––>

Galileo scale transformation Galileo scale transformation groupgroup

-comes under the principle of relativity (of scales)-

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ln L

ln ε

transition

fractal

ln ε

transitionfractal

delta special scale-relativity

Planck scale

scaleindependent

scaleindependentPlanck scale

variation of the scale dimensionvariation of the length

(Simplified case : )

Scale dependence of the length and of the Scale dependence of the length and of the effective scale dimension in special scale-effective scale dimension in special scale-

relativity (log-Lorentzian laws of scale relativity (log-Lorentzian laws of scale transformations)transformations)

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Scale dynamics

Scale laws that are solutions of second order partial differential equations in the scale space

Least action principle in scale space ––> Euler Lagrange scale equations in terms of the « djinn »

Resolution identified as « scale velocity »:

Djinn (variable scale dimension) identified with « scale time »

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ln L

ln ε

transitionfractal

ln ε

transitionfractal

delta constant "scale-force"

variation of the scale dimension

scaleindependent

scaleindependent

variation of the length

(asymptotic)

'Scale dynamics': scale dependence of the length and of the effective scale-dimension in the case of a constant 'scale-force'

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‘Scale dynamics’: scale dependence of the length and of the effective scale-dimension in the case of an harmonic oscillator ‘scale-potential’

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Scale dependence of the length and of the scale dimension in the case of a log-periodic behavior (discrete scale invariance) including a fractal / nonfractal transition.

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Foundation of Foundation of quantum quantum

mechanicsmechanicsEffets on the motion equationsEffets on the motion equations

of the of the

fractal structures internal to geodesicsfractal structures internal to geodesics

http://www.luth.obspm.fr/~luthier/nottale/

Cf: Nottale Fractal Space-Time World Scientific (1993); Célérier Nottale J. Phys. A 37, 931 (2004); 39, 12565 (2006); Nottale Célérier J. Phys. A 40, 14471 (2007)

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Fractality Discrete symmetry breaking (dt)

Infinity ofgeodesics

Fractalfluctuations

Two-valuedness (+,-)

Fluid-likedescription

Second order termin differential equations

Complex numbers

Complex covariant derivative

NON-NON-DIFFERENTIABILITYDIFFERENTIABILITY

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Road toward Schrödinger Road toward Schrödinger (1): infinity of geodesics(1): infinity of geodesics

––> generalized « fluid » approach:

Differentiable Non-differentiable

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Road toward Schrödinger (2): Road toward Schrödinger (2): ‘differentiable part’ and ‘fractal ‘differentiable part’ and ‘fractal

part’part’Minimal scale law (in terms of the space resolution):

Differential version (in terms of the time resolution):

Case of the critical fractal dimension DF = 2:

Stochastic variable:

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Road toward Schrödinger (3): Road toward Schrödinger (3): non-differentiability ––> complex non-differentiability ––> complex

numbersnumbersStandard definition of derivative

DOES NOT EXIST ANY LONGER ––> new definition

TWO definitions instead of one: they transform one in another by the reflection (dt <––> -dt )

f(t,dt) = fractal fonction (equivalence class, cf LN93)Explicit fonction of dt = scale variable (generalized « resolution »)

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Covariant derivative operatorCovariant derivative operatorClassical(differentiable)part

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Covariant derivative operator

Fundamental equation of dynamics

Change of variables (S = complex action) and integration

Generalized Schrödinger equation

FRACTAL SPACE-TIME–>QUANTUM FRACTAL SPACE-TIME–>QUANTUM MECHANICSMECHANICS

Ref: LN, 93-04, Célérier & Nottale 04-07. See also works by: Ord, El Naschie, Hermann, Pissondes, Dubois, Jumarie, Cresson, Ben Adda, Agop, et al…

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Newton

Schrödinger

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Application in Application in astrophysics: astrophysics: gravitational gravitational

structuresstructuresMacroscopic Macroscopic

Schrödinger equationSchrödinger equation

http://www.luth.obspm.fr/~luthier/nottale/

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Three representations

Geodesical (U,V) Generalized Schrödinger (P,)

Euler + continuity (P, V)

New « potential » energy:

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Gauge invariance of gravitationalSchrödinger equation

Gauge transformation of :case ofKepler potential --> dimensionless

One finds invariance under the transformation:

Provided

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n=0 n=1

n=2(2,0,0)

n=2(1,1,0)

E = (3+2n) mD

Hermite polynomials

Solutions: 3D harmonic oscillator potential 3D (constant Solutions: 3D harmonic oscillator potential 3D (constant density)density)

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Application to the Application to the formation pf planetary formation pf planetary

systemssystems

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Simulation of trajectorySimulation of trajectory

Kepler central potential GM/rState n = 3, l = m = n-1

Process:

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n=3

Solutions: Kepler potentialSolutions: Kepler potential

Generalized Laguerre polynomials

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Solar System :Solar System : inner and outer systems inner and outer systems

SI

J

S

U

N

P

m VT

M HunC

HHil

1

4

9

16

25

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rank n101 2 3 4 5 6 7 8 9

√a (obs.)

7 49

1

2

3

4

5

6

SE

N

Ref: LN 1993, Fractal space-time and microphysics (World Scientific) Chap. 7.2

New predictions

(at that time)0.043 UA/Msol 0.17 UA/Msol

55 UA

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Outer solar system:Outer solar system:Kuiper belt (SKBOs)Kuiper belt (SKBOs)

60 70 80 90 100 110 120 130

2

4

6

8

10

Semi-major axis (A.U.)

SKBO

7 8 9 10

10 20 30 40 500

2 3 4 65Rank n

1

Ref: Da Rocha Nottale 03

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Outer Solar System:Outer Solar System:Kuiper belt (SKBOs)Kuiper belt (SKBOs)

60 70 80 90 100 110 120 130

2

4

6

8

10

Semi-major axis (A.U.)

SKBO

7 8 9 10

10 20 30 40 500

2 3 4 65Rank n

1

Ref: Da Rocha Nottale 03

2003 UB 313 (« Eris »)

Validation of predicted probability peak at 55 AU

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New New planet:Sednaplanet:Sedna

2001

FP

185

Sed

na 2

003

VB

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( a / 57 UA )1/2

SK

BO

s

nex=7

PredictePredicted,AUd,AU (57)(57) 228228 513513 912912 142142

5520520522

ObserveObservedd 5757 227227 509509

Num

ber

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Solar System: Sun, solar Solar System: Sun, solar cyclecycle

If the Sun had kept its initial rotation: would then be the Kepler period,

But, like all stars of solar-type, the Sun has been subjected to an important loss of angular momentum since its formation (cf. Schatzman & Praderie, The Stars, Springer)

Wave function:

Fundamental period:

On the surface of the Sun:

(Pecker Schatzman)

Result: Observed period:11 ans

Ref: LN, Proceedings of CASYS’03, AIP Conf. Proc. 718, 68 (2004)

(equator)

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Exoplanets (data 2006)

(P / M*)^(1/3)

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Exoplanets (data 2008, N=301)

(P / M*)^(1/3)

Num

ber

Predicted probability peaks

(main peak cut)

Proba = 5 x10-7

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Exoplanets (data 2008, N=301)Main peak

Predicted (1993) fundamental level, 0.043 AU/ Msol

mer

cury

Ven

us

Ear

th

Mar

s

Cer

es

Hyg

eia

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Extrasolar planetary system:PSR B1257+12

25 2624 66 67 98 9997

10 20 30 40 50 60 70 80 90 1000 110

Period (days)

days days days

1 2 3 4 5 6 7 8

A B C

Refs: Nottale 96, 98, Da Rocha & Nottale 03

Data:Wolszczan 94, 00

Mpsr =1.4 ± 0.1 Msol --> w = (2.96 ± 0.07) x 144 km/s, i.e. 432 km/s = Keplerian velocity for Rsol

Proba < 10-5 of obtaining such an agreement by chance

Prediction of other orbits: P1=0.322 j, P2=1.958 j, P3=5.96 j

Residuals in Wolszczan’s data 00: P = 2.2 j (2.7 )

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Comparison to the inner Solar System

m V T M

Distance to the star, normalized by its mass (MPSR=1.5 Msol). n^2 law

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New comparison to the TSR prediction (improved observational data, Wolszczan et al 2003)

A B C

Base: planet C : aC = 68, nC = 8

Planet A: (aA)pred = 27.5 <--> (aA)obs = 27.503 ± 0.002

(nA)pred = 5 <--> (nA)obs = 5.00028 ± 0.00020

Planet B: (aB)pred = 52.5 <--> (aB)obs = 52.4563 ± 0.0001

(nB)pred = 7 <--> (nB)obs = 6.997 ± 0.00001

nA/nA = 5 x 10-5 Improvement by a factor 12 !

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Stars:Planetary nebulae

Da Rocha 2000, Da Rocha & Nottale 2003

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Stars:ejection and accretion

SN 1987A, deprojected angle : 41.2 ± 1.0 d° predeicted angle: (l=4, m=2): 40.89 d°

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Applications of scale Applications of scale laws in geosciences:laws in geosciences:

critical and log-periodic critical and log-periodic lawslaws

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Arctic sea ice extent decrease

Tc = 2012 --> free from ice in 2011 ! (possibly 2010: expected 1 M km2

(Minimum 15 september of each year)

Critical power lawy0-a (T-Tc)-g

2007 and 2008 values predicted before observation(Nottale 2007)

Constant rate

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Arctic sea ice extent decrease(Mean August)

Confirmation: full melting one year later (2012)

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South California earthquake rate

Log-periodic deceleration from ~1796, g=1.27

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May 2008

SichuanSeism

Date (day, May 2008)

magnitude

rate

Log-periodic

deceleration

of

replicas

Mainearthquake

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Applications in physics Applications in physics and cosmologyand cosmology

Special scale relativity --> value of strong coupling

Scale-dependent vacuum --> value of cosmological constant

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Comparison to experimental data + extrapolation by renormalization group

10 1 10-3 3

106

109

1012

1018

1027

Energy (GeV)

10

20

30

40

50

4π 2

eWZt GUT

e

0 10 20 30 40 50

l

α1

α0

α2

α3

αg

∞-1

-1

-1

-1

-1

λln ( / r )

C ( )λ

QCD

p

r0

« Bare » (infinite energy) effective electromagnetic inverse coupling

Grand unification chromodynamics and gravitational inverse couplings

Mass-coupling relations(from scale-relativisticgauge theory)

New:E = 3.2 1020 eV

Electroweakunificationscale

Predicted strongcoupling at Z scale0.1173(4)

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Comparison between theoretical prediction and

experimental value of alphas(mZ)

Date prediction

prediction

Data: PDG 1992-2006

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Value of the Value of the cosmological constantcosmological constant

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0 10 20 30 40 50 60

-140

-120

-100

-80

-60

-40

-20

0q ν

r

r

-4

-6

log (r / l )pl p L

Λ

Vac

uum

ene

rgy

dens

ity

Nottale L. 1993, Fractal Space-Time and Microphysics (World Scientific)

Nottale L., 2003, Chaos Solitons and Fractals, 16, 539. "Scale-relativistic cosmology" http://www.luth.obspm.fr/~luthier/nottale/NewCosUniv.pdf

5.3 x 10-3 eVe ?

Cosmological constant and vacuum energy density

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Cosmological constant and vacuum energy density.

Value of r0 ? Conjecture: quark-hadron + electron-electron transition during primordial universe *Largest interquark distance: ––> Compton length of effective mass of quarks in pion:

*QCD scale for 6 quarks (extrapolation):

*Classical radius of the electron–––> e-e cross section re

2

–––> Result:

= 1.362 10-56 cm-2

h2= 0.38874(12)

H0=71 ± 3 km/s.Mpc, = 0.73 ± 0.04 (Wmap…)

Predicted (LN 93): Observed:

h2= 0.40 ± 0.03

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Comparison prediction-observations

Gunn-Tinsley LN, Hubblediagram ofInfraredellipticals

LN, age problem

SNe,WMAP 3yrlensing

SNeI SNe,WMAP1yrlensing

prediction